Indestructible dynamics of torus maps

Given a $d$-dimensional torus map $F(z)=Mz+G(z)\bmod 1$, where $M$ is an integer-matrix and and $G$ is a periodic function, we find conditions on $M$ under which $F$ is semi-conjugate to a linear torus map, independently of $G$. We also find a conditions $G$ under which these semi-conjugacies can be turned into conjugacies. These conditions are satisfied by open sets of torus maps (in the $C^1$-topology) and therefore describe some asymptotic behavior of trajectories which are stable under perturbations to the map.